Abstract
We derive sufficient conditions for controllability and necessary conditions for minimum in nonsmooth optimal control problems defined by differential or functional-integral equations with isoperimetric and unilateral restrictions. We consider the cases when the controls are relaxed or chosen fromabundant sets of original (ordinary) controls (which include most, or all, of the control sets studied in the literature). We prove that, if there exist optimal strictly original controls (that is, controls that are optimal in an abundant set but not among relaxed controls), then the problem admits abnormal extremals. We also study the abnormality of the optimal strictly original controls themselves.
Similar content being viewed by others
References
Warga, J.,Optimal Control of Differential and Functional Equations, Academic Press, New York, New York, 1972.
Warga, J.,Controllability of Nondifferentiable Hereditary Processes, SIAM Journal on Control and Optimization, Vol. 16, pp. 813–831, 1978.
Warga, J.,Optimization and Controllability without Differentiability Assumptions, SIAM Journal on Control and Optimization (to appear).
Warga, J.,Necessary Conditions without Differentiability Assumptions in Unilateral Control Problems, Journal of Differential Equations, Vol. 21, pp. 25–38, 1976.
Warga, J.,Controllability and Necessary Conditions in Unilateral Problems without Differentiability Assumptions, SIAM Journal on Control and Optimization, Vol. 14, pp. 546–572, 1976.
Warga, J.,Derivate Containers, Inverse Functions, and Controllability, Calculus of Variations and Control Theory, Edited by D. L. Russell, Academic Press, New York, New York, 1976.
Warga, J.,Optimal Control of Differential and Functional Equations, Chapter 11, Nauka, Moscow, 1977 (in Russian).
Vinter, R. B., andPappas, G.,A Maximum Principle for Nonsmooth Optimal Control Problems with State Constraints, Journal of Mathematical Analysis and Applications, Vol. 89, pp. 212–232, 1982.
Warga, J.,Controllability and a Multiplier Rule for Nondifferentiable Optimization Problems, SIAM Journal on Control and Optimization, Vol. 16, pp. 803–812, 1978.
Warga, J.,Normal Control Problems Have No Minimizing Strictly Original Solutions, Bulletin of the American Mathematical Society, Vol. 77, pp. 625–628, 1971.
Warga, J.,A Second-Order Condition that Strengthens Pontryagin's Maximum Principle, Journal of Differential Equations, Vol. 28, pp. 284–307, 1978.
Warga, J.,Fat Homeomorphisms and Unbounded Derivate Containers, Journal of Mathematical Analysis and Applications, Vol. 81, pp. 545–560, 1981.
Author information
Authors and Affiliations
Additional information
Communicated by M. R. Hestenes
Dedicated to L. Cesari
This work was supported in part by the National Science Foundation under Grant No. MCS-81-02079.
Rights and permissions
About this article
Cite this article
Warga, J. Controllability, extremality, and abnormality in nonsmooth optimal control. J Optim Theory Appl 41, 239–260 (1983). https://doi.org/10.1007/BF00934445
Issue Date:
DOI: https://doi.org/10.1007/BF00934445